Finite Group Databasebeta

← C3×#70105 #11788804

Subgroups — C3×#70105 #11788804

Order 1152.

701 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5816C211yesmaximal, characteristic
2C3×#5812C211yesmaximal, characteristic, complemented
3C2×C32×#101C211yesmaximal, characteristic, complemented
4C3×#5139C212yesmaximal
5C3×#4557C212yesmaximal, complemented
6C3×#1912—31nomaximal
7#70105C311yesmaximal, characteristic, direct product factor
8C22×C32×Q8C2211yescharacteristic
9C2×C3×#235C2211yescharacteristic
10C2×C32×M16C2212yes—
11C3×#947C412yescomplemented
12C3×#946—22no—
13C3×Q8×Dic3—24no—
14C3×#944C412yescomplemented
15C3×#929—22no—
16C3×#926—22no—
17C3×#926C2211yescharacteristic
18C3×#926C2212yes—
19C32×#101—22no—
20C3×#825—24no—
21C3×#536—31no—
22C3×#519—31no—
23#5816C2×C311yescharacteristic
24#5812C2×C311yescharacteristic, complemented
25C3×#421—32no—
26C2×C3×#101—21no—
27C2×C3×#101D311yescharacteristic, complemented
28C2×C3×#101C2×C311yescharacteristic, complemented
29#5139C2×C312yes—
30C3×H(3,ℤ/4)—32no—
31#4557C2×C312yescomplemented
32C2×C32×Q8—22no—
33C2×C32×Q8—22no—
34C2×C32×Q8C2×C412yes—
35C2×C32×Q8D412yes—
36C22×C32×C4D412yes—
37C22×C32×C4C2311yescharacteristic
38C2×C3×#67—21no—
39C3×#235—22no—
40C3×#235—22no—
41C2×C3×D12C2×C411yescharacteristic
42C2×C3×C4×D3—21no—
43C2×C3×Dic6C2×C411yescharacteristic
44C32×M16—22no—
45C32×M16—22no—
46C2×C32×C8—22no—
47C3×#217—22no—
48C3×#217—21no—
49C3×(C2×D3)⋊C4—24no—
50C3×C4⋊Dic3—24no—
51C3×C4⋊Dic3—22no—
52C3×C4⋊Dic3—21no—
53C3×Dic3⋊C4—24no—
54C3×C4×Dic3—24no—
55C3×C4×Dic3C2×C412yes—
56C3×C4×Dic3D414yes—
57#1912—31noSylow
58C2×C3×#41—31no—
59C22×C3×Q8—21no—
60C22×C3×Q8C2×D311yescharacteristic
61C22×C3×Q8C22×C311yescharacteristic
62C2×#235C22×C311yescharacteristic
63C2×C3×M16—22no—
64C2×C3×M16C2×D312yes—
65C2×C3×M16C22×C312yes—
66C3×C4⋊Q8—32no—
67C3×#124—32no—
68C3×#123—62no—
69C3×C22⋊Q8—62no—
70C3×C4×Q8—64no—
71C3×#117—32no—
72C3×#117—31no—
73C3×#117—62no—
74#947C3×C412yescomplemented
75#946—22no—
76Q8×Dic3—24no—
77#944C3×C412yescomplemented
78#929—22no—
79#926—22no—
80#926C22×C311yescharacteristic
81#926C22×C312yes—
82C3×#104—64no—
83C3×#101—24no—
84C3×#101—22no—
85C3×#101—22no—
86#825—24no—
87C23×C32C2×D411yescharacteristic
88C22×C3×D3—21no—
89C32×Q8—24no—
90C32×Q8—24no—
91C2×C32×C4—22no—
92C2×C32×C4—22no—
93C2×C32×C4C22⋊C414yes—
94C2×C32×C4C22×C411yescharacteristic
95C2×C32×C4C2×D412yes—
96C2×C32×C4C2×D411yescharacteristic
97C2×C32×C4#4112yes—
98C3×#67—41no—
99C3×#67—41no—
100C2×C3×Dic3—22no—
101C2×C3×Dic3—21no—
102C2×C3×Dic3—24no—
103C3×D12—22no—
104C3×C4×D3—42no—
105C3×Dic6—22no—
106C32×C8—42no—
107#536—31no—
108#519—31no—
109#421—32no—
110C2×#101C3×D311yescharacteristic, complemented
111H(3,ℤ/4)—32no—
112C3×#41—62no—
113C3×#41—62no—
114C2×C3×Q8—22no—
115C2×C3×Q8—22no—
116C2×C3×Q8—22no—
117C2×C3×Q8—22no—
118C2×C3×Q8—22no—
119C2×C3×Q8—22no—
120C2×C3×Q8—31no—
121C2×C3×Q8—42no—
122C2×C3×Q8—42no—
123C2×C3×Q8C4×D312yes—
124C2×C3×Q8#6712yes—
125C2×C3×Q8C2×C3×C412yes—
126C2×C3×Q8C3×D412yes—
127C2×C3×D4—31no—
128C2×C3×D4—61no—
129C22×C3×C4—22no—
130C22×C3×C4—21no—
131C22×C3×C4—61no—
132C22×C3×C4D1212yes—
133C22×C3×C4C3×D412yes—
134C22×C3×C4C22×D311yescharacteristic
135C22×C3×C4C23×C311yescharacteristic
136C2×#67—21no—
137#235—22no—
138#235—22no—
139C2×D12C2×C3×C411yescharacteristic
140C2×C4×D3—21no—
141C2×Dic6C2×C3×C411yescharacteristic
142C3×M16—24no—
143C3×M16—22no—
144C3×M16—22no—
145C3×M16—22no—
146C3×M16—22no—
147C3×M16—42no—
148C2×C3×C8—22no—
149C2×C3×C8—22no—
150C2×C3×C8—42no—
151C3×C4⋊C4—64no—
152C3×C4⋊C4—64no—
153C3×C4⋊C4—62no—
154C3×C4⋊C4—61no—
155C3×C22⋊C4—62no—
156C3×C22⋊C4—61no—
157C3×C22⋊C4—64no—
158C3×C42—32no—
159C3×C42—34no—
160C3×C42—64no—
161#217—22no—
162#217—21no—
163(C2×D3)⋊C4—24no—
164C4⋊Dic3—24no—
165C4⋊Dic3—22no—
166C4⋊Dic3—21no—
167Dic3⋊C4—24no—
168C4×Dic3—24no—
169C4×Dic3C2×C3×C412yes—
170C4×Dic3C3×D414yes—
171C22×C32—41no—
172C22×C32C2×C22⋊C411yescharacteristic
173C22×C32#12011yescharacteristic
174C22×C32#12311yescharacteristic
175C2×C3×D3—42no—
176C2×C3×D3—41no—
177C32×C4—24no—
178C32×C4C4×D412yes—
179C32×C4#12112yes—
180C3×Dic3—44no—
181C3×Dic3—42no—
182C3×Dic3—41no—
183C2×#41—31no—
184C22×Q8C2×C3×D311yescharacteristic
185C2×M16C2×C3×D312yes—
186C4⋊Q8—32no—
187#124—32no—
188#123—62no—
189C22⋊Q8—62no—
190C4×Q8—64no—
191#117—32no—
192#117—31no—
193#117—62no—
194#104—64no—
195#101—22no—
196C23×C3—21no—
197C23×C3—61no—
198C23×C3C2×D1211yescharacteristic
199C23×C3C2×C3×D411yescharacteristic
200C22×D3—21no—
201C3×Q8—24no—
202C3×Q8—24no—
203C3×Q8—24no—
204C3×Q8—24no—
205C3×Q8—44no—
206C3×Q8—44no—
207C3×Q8—62no—
208C3×D4—62no—
209C3×D4—121no—
210C3×D4—121no—
211C2×C3×C4—22no—
212C2×C3×C4—22no—
213C2×C3×C4—22no—
214C2×C3×C4—22no—
215C2×C3×C4—24no—
216C2×C3×C4—21no—
217C2×C3×C4—22no—
218C2×C3×C4—22no—
219C2×C3×C4—21no—
220C2×C3×C4—42no—
221C2×C3×C4—42no—
222C2×C3×C4—64no—
223C2×C3×C4—62no—
224C2×C3×C4—61no—
225C2×C3×C4—122no—
226C2×C3×C4(C2×D3)⋊C414yes—
227C2×C3×C4C3×C22⋊C414yes—
228C2×C3×C4C2×C4×D311yescharacteristic
229C2×C3×C4D3×D412yes—
230C2×C3×C4#23712yes—
231C2×C3×C4C2×#6711yescharacteristic
232C2×C3×C4C22×C3×C411yescharacteristic
233C2×C3×C4C2×C3×D412yes—
234C2×C3×C4C2×C3×D411yescharacteristic
235C2×C3×C4C3×#4112yes—
236#67—41no—
237#67—41no—
238C2×Dic3—22no—
239C2×Dic3—21no—
240C2×Dic3—24no—
241D12—22no—
242C4×D3—42no—
243Dic6—22no—
244C3×C8—44no—
245C3×C8—42no—
246C3×C8—42no—
247C2×C32—21no—
248C2×C32—21no—
249C2×C32—21no—
250C2×C32#38611yescharacteristic
251C3×D3—81no—
252#41—62no—
253#41—62no—
254C2×Q8—22no—
255C2×Q8—22no—
256C2×Q8—31no—
257C2×Q8C3×C4×D312yes—
258C2×Q8C3×#6712yes—
259C2×D4—31no—
260C2×D4—61no—
261C22×C4—61no—
262C22×C4C3×D1212yes—
263C22×C4C22×C3×D311yescharacteristic
264M16—22no—
265M16—22no—
266C2×C8—22no—
267C4⋊C4—64no—
268C4⋊C4—64no—
269C4⋊C4—62no—
270C4⋊C4—61no—
271C22⋊C4—62no—
272C22⋊C4—61no—
273C22⋊C4—64no—
274C42—32no—
275C42—34no—
276C42—64no—
277C22×C3—21no—
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—41no—
281C22×C3—41no—
282C22×C3—42no—
283C22×C3—122no—
284C22×C3—121no—
285C22×C3#88211yescharacteristic
286C22×C3#88611yescharacteristic
287C22×C3C2×(C2×D3)⋊C411yescharacteristic
288C22×C3C2×C3×C22⋊C411yescharacteristic
289C22×C3C3×#12011yescharacteristic
290C22×C3C3×#12311yescharacteristic
291C2×D3—42no—
292C2×D3—41no—
293C3×C4—24no—
294C3×C4—24no—
295C3×C4—22no—
296C3×C4—22no—
297C3×C4—44no—
298C3×C4—124no—
299C3×C4—122no—
300C3×C4—121no—
301C3×C4#67⋊C412yes—
302C3×C4#93912yes—
303C3×C4C3×C4×D412yes—
304C3×C4C3×#12112yes—
305Dic3—44no—
306Dic3—42no—
307Dic3—41no—
308C32#191211yescharacteristic, complemented, Sylow
309C23—61no—
310C23C2×C3×D1211yescharacteristic
311Q8—24no—
312Q8—24no—
313Q8—62no—
314D4—62no—
315D4—121no—
316D4—121no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—64no—
320C2×C4—62no—
321C2×C4—61no—
322C2×C4—122no—
323C2×C4C3×(C2×D3)⋊C414yes—
324C2×C4C2×C3×C4×D311yescharacteristic
325C2×C4C3×D3×D412yes—
326C2×C4C3×#23712yes—
327C2×C4C2×C3×#6711yescharacteristic
328C8—42no—
329C2×C3—22no—
330C2×C3—21no—
331C2×C3—21no—
332C2×C3—21no—
333C2×C3—21no—
334C2×C3—21no—
335C2×C3—21no—
336C2×C3—21no—
337C2×C3—41no—
338C2×C3—41no—
339C2×C3—241no—
340C2×C3#495811yescharacteristic
341C2×C3C3×#38611yescharacteristic
342D3—81no—
343C22—41no—
344C22—122no—
345C22—121no—
346C22C3×#88211yescharacteristic
347C22C3×#88611yescharacteristic
348C22C2×C3×(C2×D3)⋊C411yescharacteristic
349C4—24no—
350C4—124no—
351C4—122no—
352C4—121no—
353C4C3×#67⋊C412yes—
354C4C3×#93912yes—
355C3—21no—
356C3#7010511yescharacteristic, direct product factor
357C3C3×#191211yescharacteristic, complemented
358C2—21no—
359C2—21no—
360C2—21no—
361C2—241no—
362C2C3×#495811yescharacteristic
subgroupG/H
centerC2×C3#4958
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC2×C32×#101C2
socleC2×C32#386
prime radicalC2×C3×#235C22
cosocleC22×C3×C4C23×C3
hypercenterC2×C3×#101D3
hypocenterC3C3×#1912
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C2×C3×#101

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C3×#5812C2semidirect
3. C2×C32×#101C2semidirect
5. C3×#4557C2semidirect
7. #70105C3direct (N × H)
11. C3×#947C4semidirect
14. C3×#944C4semidirect
24. #5812C2×C3semidirect
27. C2×C3×#101D3semidirect
28. C2×C3×#101C2×C3semidirect
31. #4557C2×C3semidirect
74. #947C3×C4semidirect
77. #944C3×C4semidirect
110. C2×#101C3×D3semidirect
308. C32#1912semidirect
356. C3#70105direct (N × H)
357. C3C3×#1912semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C3×#58162
C3×#58122
C2×C32×#1012
C3×#51392
C3×#45572
C3×#19123
#701053